We are given the sequence beginning 2, 6, 5x-2. We are asked to find the value of x to make the sequence geometric.
A geometric sequence is of the form `a,ar,ar^2,ar^3,...,ar^n,...` where r is the common ratio.
1. Since the first term is 2 and the second term is 6, we know that the common ratio r=3. Then the third term will be 6(3)=18. So 5x-2=18 implies that x=4.
2. If the three terms form a geometric sequence, then the second term is the geometric mean between the first and third terms. Thus,
`2/6=6/(5x-2) ==> 2(5x-2)=36 ==> 10x=40 ==> x=4.`
3. The third term of a geometric series is `ar^2` where a is the first term (here 2) and r the common ratio (here r=3), so the third term is `2(3)^2=18` so then 5x-2=18 and x=4 as above.